

 Under the notion that with aggregation or weak stabilization \(\beta \lesssim h^{-1}\) , the continuity follows directly from the Cauchy-Schwarz inequality together with the observation that \(\partial _n v \in L^2(\partial \Omega _D)\)  for \(v \in V_{h,\bigstar } \oplus H^2(\Omega )\)  and therefore \(\Vert  \partial _n v \Vert _{\partial \Omega _D}\)  is well defined. For the coercivity we use the element-wise trace inverse inequality [1]}
\(h \Vert  \partial _n v \Vert ^2_{T \cap \partial \Omega _D} \lesssim \Vert  \nabla v \Vert ^2_T \qquad v \in \mathcal {P}_p(T)\) 
